The logarithmic Sarnak conjecture for ergodic weights

TL;DR

This paper proves the logarithmically averaged Sarnak conjecture for a broad class of dynamical systems, revealing structural properties of systems linked to the M"obius and Liouville functions, and showing the Liouville function's super-linear block growth.

Contribution

It establishes the conjecture for uniquely ergodic zero-entropy systems and uncovers their structural composition involving nilsystems and Bernoulli systems.

Findings

The M"obius and Liouville systems have no irrational spectrum.

Liouville function exhibits super-linear block growth.

Structural analysis connects ergodic theory with number theory via Tao's identities.

Abstract

The M\"obius disjointness conjecture of Sarnak states that the M\"obius function does not correlate with any bounded sequence of complex numbers arising from a topological dynamical system with zero topological entropy. We verify the logarithmically averaged variant of this conjecture for a large class of systems, which includes all uniquely ergodic systems with zero entropy. One consequence of our results is that the Liouville function has super-linear block growth. Our proof uses a disjointness argument and the key ingredient is a structural result for measure preserving systems naturally associated with the M\"obius and the Liouville function. We prove that such systems have no irrational spectrum and their building blocks are infinite-step nilsystems and Bernoulli systems. To establish this structural result we make a connection with a problem of purely ergodic nature via some identities recently obtained by Tao. In addition to an ergodic structural result of Host and Kra, our analysis is guided by the notion of strong stationarity which was introduced by Furstenberg and Katznelson in the early 90's and naturally plays a central role in the structural analysis of measure preserving systems associated with multiplicative functions.